Al-Gwaiz M.A. Sturm-Liouville Theory and its Applications

London: Springer-Verlag, 2008. — 270 p.This book is based on lecture notes which I have used over a number of years to teach a course on mathematical methods to senior undergraduate students of mathematics at King Saud University. The course is offered here as a prerequisite for taking partial differential equations in the final (fourth) year of the undergraduate program. In Chapter 1 the stage is set by defining the inner product space of square integrable functions. The difficulty with building Fourier analysis on the Sturm–Liouville theory is that the latter is deeply rooted in functional analysis, in particular the spectral theory of compact operators, which is beyond the scope of an undergraduate treatment such as this. We need a simpler proof of the existence and completeness of the eigenfunctions. In the case of the regular Sturm–Liouville problem, this is achieved in Chapter 2 by invoking the existence theorem for linear differential equations to construct Green’s function for the Sturm–Liouville operator, and then using the Ascoli–Arzela theorem to arrive at the desired conclusions. Chapters 3 through 5 present special applications of the Sturm–Liouville theory. In Chapters 4 and 5 we discuss some of the more important examples of a singular Sturm–Liouville problem. Chapters 6 and 7 on the Fourier and Laplace transformations are not really part of the Sturm–Liouville theory, but are included here as extensions of the Fourier series method for representing functions. This book follows a strict mathematical style of presentation, but the subject is important for students of science and engineering.Inner Product Space
The Sturm–Liouville Theory
Fourier Series
Orthogonal Polynomials
Bessel Functions
The Fourier Transformation
The Laplace Transformation
Solutions to Selected ExercisesNotation